50 
MR. AIRY ON THE LAWS OF INDIVIDUAL TIDES 
If we make phase + 58°*30' = p, this becomes 
0*900 -f- 0*898 . sin p + 0*194 . sin (2 p — 1°) 4- 0*191 . (sin 3 p — 68° 8') 
— 0*034 . sin (4 p — 1 74° 1 7') . 
The theory of waves, so far as it has yet gone, fails to explain the form of this ex- 
pression. It is not reconcileable with that obtained on the supposition that the extent 
of vertical oscillation bears a sensible proportion to the depth, either when the length 
of the channel is indefinite, or when the channel is interrupted by a barrier*. The 
latter supposition is that which would seem to represent most exactly the circum- 
stances of Southampton. It is to be remarked, however, that the investigations which 
I have cited suppose the section of the channel to be rectangular ; a supposition 
which accords little with the state of the channel of the Southampton water, as shown 
by the extensive banks of mud discovered at low water there. The investigations 
also exclude friction. 
The following consideration is suggested by the form of the expression at which 
we have arrived for the converted depression. If the tide were such as we suppose 
the sea-tide to be (that is, if its depression were expressed by a single sine or cosine), 
and if (as above) the phase were measured from low water, then the converted de- 
pression would be expressed by 
A' -{- B' cos phase, 
or A' + B' sin (phase + 90°). 
From this we might be led to conceive that in all cases the argument of the principal 
term expressing the depression would be (phase + 90°), the phase being a quantity 
which commences from low water. But we find that it does in this instance depend 
on (phase + 58° 30'), or on (phase + 90° — 31° 30'), differing from the former by 31° 30', 
which corresponds to one hour of time nearly. Now there seems to be no reason (ex- 
cept the convenience of mariners) why cotidal lines and speculations on the progress 
of the tide should be connected with high water rather than with low water, or any 
other phase of tide ; the only thing with which, in a scientific view, they can properly 
be connected (as it appears to me), is the epoch of the argument of the principal term 
in the formula for depression. But, in the instance of Southampton, the time of 
actual low water differs from the time of greatest depression given by that term, by 
one hour ; the times of high water differ about forty minutes. If then the peculiarity 
of the Southampton tide has been created in part by the shallowness of the English 
channel, and exists on the coast (we have no accurate information whether this is 
true or not), then the position of the cotidal line determined by the high water is 
erroneous, with reference to the views above mentioned, to a considerable extent. 
Another consideration suggested by the same expression is this. The mean of the 
depressions at high-water and at low-water is T000. But the constant term which 
enters into the general formula for the depression is 0*900. Now I conceive the latter 
to be the number which truly represents the mean level of the water. In all cases, 
* See Philosophical Transactions, 1842, p. 6. 
